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We can sums over multiplicative Dirichlet characters $\sum\limits_{N<n<M}\chi(n)$ using the Polya Vinogradov inequality by $\ll \sqrt{p}\log\,p .$ Is there a similar estimate when we sum over primitive roots modulo q, where q is a fixed integer: $$\sum\limits_{\substack{N<n<M\\ (n,q)=1}}e^{2\pi i \frac{n}{q}}$$

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    $\begingroup$ No; consider what happens with $N=1$ and $M=q/100$. $\endgroup$
    – Lucia
    Commented Mar 8, 2018 at 3:04
  • $\begingroup$ Hi, I do not understand what you mean. $\endgroup$
    – Claus
    Commented Mar 8, 2018 at 3:25
  • $\begingroup$ You say "primitive roots", but your sum is over all reduced residue classes; is that what you mean? $\endgroup$ Commented Mar 8, 2018 at 7:42
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    $\begingroup$ Lucia meant that for $0<\frac{n}{q}<1/100$ we have $\mathrm{Re}\,e^{\frac{2\pi i n}{q}}=\cos\left(\frac{2\pi i n}{q}\right)>\cos\left(\frac{\pi}{50}\right)$ and thus for $M=q/100, N=1$ the real part of your sum is at least $\cos\left(\frac{\pi}{50}\right)\#\{N<n<M: (n,q)=1\}$, i.e. you don't have any nontrivial estimate (up to some constant). $\endgroup$ Commented Mar 8, 2018 at 8:02
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    $\begingroup$ Also, unlike in the case of Dirichlet characters, your sum may be not bounded uniformly in $M$ and $N$, because the "complete" sum (with $N=0, M=q$) equals $\mu(q)$, which can be nonzero $\endgroup$ Commented Mar 8, 2018 at 8:43

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The standard approach would use the Möbius function: \begin{align*} \sum_{\substack{N<n<M\\ (n,q)=1}}e^{2\pi i n/q} &= \sum_{N<n<M}e^{2\pi i n/q} \sum_{d\mid(n,q)} \mu(d) \\ &= \sum_{d\mid q} \mu(d) \sum_{\substack{N<n<M \\ d\mid n}}e^{2\pi i n/q} \\ &= \sum_{d\mid q} \mu(d) \sum_{N/d<m<M/d}e^{2\pi i m/(q/d)}. \end{align*} The inner sum is now a geometric series which can be evaluated exactly or approximated in standard ways....

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